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Illustration of velocity estimation based on cross-correlation at different shifts. Red and blue dots represent a feature that has moved (1 unit in each direction) between two acquisitions (red being the first scene and blue being the second). WC and WI are correlation window and interrogation window sizes respectively. a) No shift (x=0 & y=0), represented by the 0,0 element in the cross-correlation matrix shown in d. b) Shift of (x=1 & y=-1) cross-correlation value represented by the 1,-1 element in the crosscorrelation matrix in d. c) Same as b but with shift of (x=2 & y=-2), cross-correlation value represented in 2,-2 element in d. d) Cross-correlation matrix with colors in each grid representing cross correlation function, rN (red is higher). Offset of this peak from origin (0, 0) in the correlation matrix is an estimation of displacements in the x and y directions.

Several studies (e.g. Huang et al., 1997; Meunier and Leweke, 2003; Raffel et al., 2007) have discussed errors in PIV-estimated displacement fields. For use with TLS data, the most important error sources are the size of the grid GR (Figure 1.1), the magnitude of displacement gradients, and the relative window sizes WI and WC with respect to maximum displacements in the correlation window. Because of the high precision of modern laser scanners, generally less than 1 cm even in the case of complex topography (Schürch et al., 2011) and the relatively small displacement gradients (with respect to fluid flows) expected for non-catastrophically deforming landslides, we expect that the sizes of WI and WC in the first iteration will most strongly influence PIV results.

To choose these parameters most appropriately, we adhere to the following criteria (Hu et al., 1998; Meunier et al., 2004):

1) WI must be of sufficient size so that there are enough pixels with unique values to estimate a cross-correlation function. In PIV, the minimum number of particles has to be greater than four but the result will be better if 10-20 particles are visible (Meunier and

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2) To ensure that more than two thirds of the particles remain in the interrogation window, WI must be more than 3 times greater than the maximum displacement, dmax in the correlation window (Raffel et al., 2007).

3) WI must also be small enough so that it encompasses a close approximation to a homogeneous displacement field. To ensure this, the difference in the displacement magnitude in the correlation window between a group of pixels must be smaller than GR.

For higher displacement gradients, a blurring of the image with a Gaussian function is recommended (Meunier and Leweke, 2003).

4) By definition, WC must be smaller than WI (Figure 1.1) and it must be greater than two times dmax (Hu et al., 1998). Smaller WC will violate the sampling criteria (Nyquist theorem) and cause the measurement to be aliased.

5) Because the displacement gradients are smaller in second and higher order iterations, both WI and WC parameters have less influence on the result. In the higher order iterations, WI needs to have enough non-unique data pixels. In these cases, WC can be very small; even a couple of pixels may suffice.

1.4 Synthetic Examples To better understand the effect of TLS data acquisition parameters, such as laser spot spacing, on PIV-estimated displacement fields, we performed a series of tests using synthetic data analogous to TLS point cloud data acquired from a deforming ground surface of random elevations. The synthetic point clouds were created with spot spacing ranging from 10 cm to 2 m by 10 cm increments (spot-spacing was held constant for each test) and a displacement signal was applied (maximum displacement of 1 m) in the y direction to each point cloud (Figure 1.2). Images were formed with grid size GR = 0.05, 0.1, and 0.2 m using an algorithm that assigned to each node the median Z value in a square grid cell with length, GR. The grid node for cells devoid of points (particularly possible when GR spot spacing) was assigned a null value. We then applied the PIV method to estimate the displacement field with correlation parameters selected using the criteria previously described. The results show that for each value of GR, smaller spot spacing is coincident with a monotonic decrease in root mean squared (RMS) values (Figure 1.2b) between the imposed synthetic signal and the PIV-estimated value (Figure 1.2d,f). This is because denser spot-spacing yields a sharper correlation peak and a commensurately more accurate displacement estimate. Additionally, smaller GR also leads to smaller RMS. This is a function of positional error from the gridding process: for each gridded Z-value, the positional error can be as large as

0.5 GR and so accuracy increases with decreasing GR. Although by no means an exhaustive suite of synthetic experiments (the myriad displacement fields and geometries of landslides preclude such an exercise) these synthetic results demonstrate some of the first-order controls on PIV performance and demonstrate that the method can estimate a displacement field to better than 5% relative error.

1.5 Application: Cleveland Corral Landslide The Cleveland Corral landslide (CCL) in California’s Sierra Nevada Mountains is ~ 450 m long, 25-70 m wide (Figure 1.3), and usually moves after winter rainfall exceeds the mean annual rainfall level; the slide has moved multiple times since the mid-1990s (Reid et al., 2003). The CCL is one of 600 mapped landslides along a 24 km stretch of Highway 50 in California parallel to the south fork of the American River (Spittler and Wagner, 1998) and it lies within 3 km of two large landslides that failed catastrophically and blocked the highway for weeks in 1997; one of these transformed into a debris flow. The CCL has been monitored since 1997 using repeat high-precision GPS ground surveys and in situ sensors including extensometers, geophones, rain gages, and sub-surface pressure transducers (Reid et al., 2003). Two shallow seismic refraction Figure 1.2.

An illustration of the PIV method applied to synthetically produced point cloud data. a) Magnitude of the synthetic displacement signal applied in y-direction. b) Mean of RMS error as a function of spot spacing using different image resolutions, GR. c) Example of PIV estimated velocity with GR = 0.05 m and spot spacing of 0.1 m d) Residuals computed for example in (c). e) PIV estimated velocity with GR = 0.1 m and spot spacing of 1 m. f) Residuals computed for example in (e).

Figure 1.3.

a) Location of the Cleveland Corral Landslide (red box) in the Sierra Nevada Mountains, California, USA. b) Google Earth optical image and areas scanned in 2005-2007 (black box) and 2010 (red box) with corresponding scan locations (stars) c) Topographic map of the area. Landslide surface features are adapted from Reid et al. (2003) d) Shaded relief using 50 cm DEM from TLS data in 2010.

High relief and no data (black) elucidates shadow forming objects mainly vegetation. Boxes outline areas selected for PIV analysis of 2005-2007 (black) and 2010 (red) scans. All TLS data are referenced to the UTM WGS84 coordinate system, and the origin is located at (724000, 4295000) to avoid large numbers in the axes.

surveys and borehole measurements indicate that the active landslide occurs in colluvium and older landslide material that varies in thickness from 5-10 m and that the principal slip surface lies just above schist bedrock (Reid et al., 2003). When active, the CCL exhibits a broad spectrum of movement style ranging from slow-moving blocks with measured displacements up to meters per year to more rapidly moving small debris flows originating from the slide margins.

1.5.1 June 2005 - January 2007 We acquired TLS data sets in June 2005 and January 2007 (Figures 4, 5) using an Optech Ilrisd TLS. Each TLS survey consisted of scans from two locations close to the landslide toe with target distances ranging from 10 to 200 m and spot spacing ranging from 1 to 5 cm. Because the scans were conducted from nearby, and at an oblique angle to, the toe, the resulting point clouds contain large shadows devoid of measurements (Figure 1.3,1.4). The point-cloud data from each acquisition were aligned to one another by masking potentially moving areas and aligning following the iterative closest point (ICP) routine in Polyworks 10.1 software. For comparison with mapped ground features, the aligned data were then transformed into a UTM projection by point-to-point surface matching (RMS error ~ 2 m) with a 0.5 m DEM of the area prepared using aerial photographs from 2007.

Figure 1.4.

TLS point cloud data and area of interest for PIV analysis. Grey area represents no data. a,b) Data density (number of data points per square meter) and distribution of data acquired in June 2005 January 2007 respectively. c) Apparent displacement of features as indicated by arrows in the point cloud data from the boxes in a and b. d,e) Data density acquired in January 2010 and May 2010. f) Point cloud data for the box in d and e. Arrow highlights the offset of the centroids of the points from the circled feature.

Figure 1.5.

PIV estimation of synthetic displacement applied to the June 2005 point cloud data. a) Magnitude of the synthetic signal varying smoothly to a maximum of 4 m in negative y direction (black dots are June 2005 point cloud). b) PIV recovered displacement in y direction (colors) and total estimated displacements (vectors). c) Residuals (imposed synthetic signal - computed displacement) in the x direction (colored contours) with vectors showing total residuals. d) Same as c), except in y direction.

The data are non-uniformly spaced (Figure 1.4a,b,c) with density ranging from many points per square centimeter to regions with no data. Inspection of identifiable features common to both scans indicates horizontal displacements as large as 4 meters (Figure 1.4c). To find the best PIV parameters for this data set, we introduced a known synthetic displacement pattern into the 2005 point cloud with a maximum value of 4m. Using the parameter-choice criteria described above in section 3 (GR, WI, and WC of 0.2 m, 12.8 m, and 8.8 m, respectively), the residual displacement values (between known synthetic and PIV determined) were less than 5% relative error although they display a systematic spatial bias of +/- ~ 0.2 m (Figure 1.5). The bias is most likely related to the poor sampling density at y-positions greater than 120 m. Nonetheless, the PIV-derived displacement field reproduces much of the character of the synthetic input field and so we proceeded further with our analysis using the +/- ~0.2 m value as an interpretation threshold scale.

The resultant displacement field shown in Figure 1.6 is almost entirely confined to the previously mapped landslide boundary (Reid et al., 2003). The maximum horizontal displacement magnitude is ~ 5 m with most vectors oriented downhill. There are four distinct displacement maxima ranging in areal dimension from 20 to 50 m. The upslope-most maximum is correlated with previously mapped tension cracks and the downslope-most maximum is correlated with thrust faults previously mapped by Reid et al. (2003) at the landslide toe. Patterns within each maximum display contraction in the lower part and stretching in the upper part, in agreement with previous observations that large slow-moving landslides often exhibit multiple areas of extension and contraction (Baum et al., 1998; Wang et al., 2010).

To estimate the uncertainties, although general PIV-related errors are formally described in the literature (Huang et al., 1997; Westerweel, 1997), here, we derive empirical, repeatability-based error estimates by exploiting the shape of each displacement estimate’s cross-correlation function. For each displacement estimation, ij, we first calculate the width, ij, of a twodimensional Gaussian function fit to the corresponding cross-correlation matrix. Assuming that ij is proportional to ij allows us to write

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where k is a proportionality constant common to all measurements and associated with factors such as the point distribution, data density, gridding resolution, and the choice of control parameters. Because motion is expected to be zero in stable areas, any non-zero PIV estimate from the stable sites (stable slope outside the landslide) will be the error, ij. Using a large number of stable sites, therefore, allows us to estimate the proportionality constants kx and ky in the xy plane. For non-stable areas, we use the kx, ky values from the stable areas and the estimated  to find ε and derive error ellipses (Strang and Borre, 1997) (Figure 1.6). The error estimated in this way may not include errors associated with potentially high displacement gradients, but this is mitigated by the window deformation method described above (Meunier and Leweke, 2003).

Figure 1.6.

PIV estimated total displacement field and vectors (black) with error ellipses (95% significance) of CCL between June 2005 and January 2007. GPS horizontal-displacement vectors (red) and displacement vectors of features identifiable in the point cloud data (white) are plotted using the same scale as the PIV vectors. Landslide surface features (scarps, thrusts, and boundaries) are adapted from Reid et al., (2003).

1.5.2 January - May 2010 During the winter and spring of 2010, part of the toe of the Cleveland Corral landslide was active. We surveyed the slide (Figure 1.3) in January 2010 and again in May 2010 using an Optech Ilris-3d TLS. The instrument was located ~1 km across the river valley from the landslide and laser spot spacing during the surveys was 10-12 cm (Figure 1.4, 1.7). In contrast to the previous scans, the laser view was oriented at a higher angle to the landslide surface and provided good coverage of the entire slide. We aligned the data sets to one another as described above.

As with the previous example, to find the best PIV parameters for these data, we first introduced a known synthetic displacement pattern into the January point cloud with a maximum horizontal displacement of 0.5 m (Figure 1.7). This synthetic maximum is similar to the observed maximum derived from manual inspection of features displaced between the two data sets. Using the parameter-choice criteria described above in section 3 (GR, WI, and WC of 0.04 m, 2.5 m and

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